Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x³−4x²+2; between 0 and 1

Verified step by step guidance

1. Recall the Intermediate Value Theorem (IVT), which states that if a function \(f\) is continuous on a closed interval \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs, then there exists at least one \(c\) in \((a, b)\) such that \(f(c) = 0\).
2. Identify the function and the interval: here, \(f(x) = x^{3} - 4x^{2} + 2\) and the interval is \([0, 1]\).
3. Evaluate the function at the endpoints of the interval: calculate \(f(0)\) and \(f(1)\).
4. Check the signs of \(f(0)\) and \(f(1)\): if one is positive and the other is negative, then by the IVT, there is at least one root between 0 and 1.
5. Conclude that since \(f\) is a polynomial (and thus continuous everywhere) and the function values at 0 and 1 have opposite signs, there must be a real zero of \(f(x)\) between 0 and 1.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes values f(a) and f(b) at each end, then it takes any value between f(a) and f(b) at some point within the interval. This theorem is used to prove the existence of roots by showing the function changes sign.

Continuity of Polynomial Functions

Polynomial functions are continuous everywhere on the real number line, meaning there are no breaks, jumps, or holes in their graphs. This property ensures that the Intermediate Value Theorem can be applied to any interval when dealing with polynomials.

Evaluating Function Values at Given Points

To apply the Intermediate Value Theorem, you must calculate the function's values at the endpoints of the interval. If the function values have opposite signs, it indicates the function crosses zero somewhere between those points, confirming the existence of a real root.